Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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1046 Part F Quantum Optics

There is signiﬁcant interest in developing X-ray lasers

for applications in lithography and medical imaging,

but so far the lack of reliable X-ray optics for the laser

cavities has limited the technology.

71.5.2 Nuclear Pumped Lasers

These are gas lasers excited by high energy charged par-

ticles or gamma rays resulting from nuclear reactions.

Either nuclear reactors or nuclear explosives are used

as pump sources. These can operate in either a pulsed

or cw mode and produce emission covering the spectral

range from the UV through the IR [71.4]. Typical gases

range from Xe

∗

at 170 nm to CO at 5.4 µm. Other gases

that have been used include argon and nitrogen. Typical

pulsed outputs produce 10 ns pulses with energies from

2×10

−7

J to 3 J. There are signiﬁcant problems with ra-

diation damage of the laser components in these systems.

71.5.3 Free Electron Lasers

These lasers are based on a high-energy beam of elec-

trons in a spatially varying magnetic ﬁeld. The varying

ﬁeld causes the electrons to oscillate, and thus to emit

radiation at the oscillation frequency. The stimulated

emission produced under these conditions provides the

laser output. Since the electrons are making transitions

between continuum states instead of discrete states,

these systems can give high power output over the entire

spectral range from the VUV to the far IR [71.35, 36].

Powers as high as 1 GW and efﬁciencies as high as 35%

have been obtained. Beam spread is controlled by the

use of tapered instead of uniform wigglers. Both cw and

picosecond pulsed emission can be obtained.

There are three types of free electron laser conﬁgu-

rations [71.35,36]. The ﬁrst is a master oscillator power

ampliﬁer (MOPA), in which an electron beam is injected

into a wiggler in synchronism with the signal to be am-

pliﬁed. The external radiation source to drive the ampli-

ﬁer is a master oscillator such as a conventional laser sys-

tem. This is a single pass, high gain system. The second

conﬁguration is an oscillator. This is designed with re-

ﬂection at the ends of the wiggler so that the signal makes

multiple passes in the cavity. This can operate with low

gain. Since it ampliﬁes spontaneous noise, no injected

signal is necessary. The third conﬁguration is a super-

radiant ampliﬁer. In this conﬁguration, shot noise is am-

pliﬁed over a single pass through the wiggler. These sys-

tems require high current accelerators to drive them. Be-

cause their operation is based on broad band shot noise,

super-radiant ampliﬁer radiation has a broader band than

radiation from a master oscillator power ampliﬁer.

Different accelerator conﬁgurations can be used to

produce the electron beam, storage rings, induction

linacs, pulse line accelerators, etc. [71.35, 36]. These

give different beam properties such as quality, current,

and energy. They also each give a limited range of

wavelengths and temporal structure of the output.

Advances have been made recently in designing

smaller and less complex free electron lasers [71.35,36].

As this trend continues, these systems will ﬁnd important

applications in medicine and industry.

71.6 Recent Developments

Over the past eight years, the designs of all types of

lasers have continued to evolve, driven by applications

requirements for lasers with speciﬁc operating param-

eters. Some of the major advances in this time period

are summarized here.

The requirements for laser outputs of several kilo-

watts with near-diffraction-limited beam quality in the

infrared wavelength region has lead to improved designs

of CO

lasers. This includes a diffusion cooled, annular

discharge design with free-space propagation instead of

waveguiding [71.37].

The push for solid state lasers in the ultraviolet has

led to recent progress in cerium-doped ﬂuoride crys-

tal lasers that provides direct laser emission tunable

in the 280 to 330 nm spectral region. Using host crys-

tals such as LiCaAlF

, LiSrAlF

or LiLuF

has helped

to overcome the problems with excited state absorp-

tion and color center formation that has been a major

problem with more common hosts such as YAG and

YLF crystals [71.38]. These lasers are pumped by ei-

ther excimer lasers or frequency-quadrupled Nd:YAG

lasers and have produced up to 60 mJ per pulse and 60%

slope efﬁciency. The output can be pulse compressed

to 115 fs.

Thin-disk solid state laser conﬁgurations using rare

earth doped crystals and semiconductor saturable ab-

sorber mirrors for passive mode-locking have produced

ultrashort (femtosecond) pulse trains with high average

powers [71.39]. Yb:YAG thin disks reduce the problem

of thermal lensing and have achieved output powers of

up to 100 W in cw mode locked operation with near-

diffraction-limited beam quality [71.40]. To achieve

Part F 71.6

Types of Lasers 71.6 Recent Developments 1047

shorter pulses, a thin-disk Yb:KYW laser has obtained

22 W with 240 fs pulses [71.41].

Advances continue to be made in the materials for

solid state organic dye lasers. The use of polymer mater-

ials, organic-inorganic matrices, and nanoparticles has

provided advances in this type of laser [71.42–46]. Use

of semiconductor excitation and electrical excitation

may play important roles in the future of solid state

organic dye lasers.

Commercially available solid state Raman lasers

have been developed for frequency shifting to a wide

variety of wavelengths and for pulse compression to the

0.1–1 ns region [71.47–49]. Both internal and external

cavity designs have been demonstrated. New mater-

ials, such as KGd(WO

)

and KY(WO

)

, have been

used for Raman lasers along with Ba(NO

)

, which has

excellent properties for this application. An external-

resonator Raman laser using Ba(NO

)

has reached

1.3 W of power [71.50].

The major advancement in ﬁber lasers has been in

cladding geometry to allow for higher powers. Double

clad ﬁber lasers operating in single transverse modes

have exceeded 100 W of output for four-level systems

and over 1 W for three-level systems [71.51]. These in-

volve a variety of geometric shapes of cladding. A new

breakthrough in ﬁber delivery systems that will impact

the future of ﬁber lasers is the use of holey ﬁbers (pho-

tonic crystal ﬁbers) [71.52]. These are glass ﬁbers that

have a periodic array of air holes running their en-

tire length. These ﬁbers can be engineered to produce

a photonic bandgap and allow for dispersion control

and minimized nonlinear effects compared to standard

ﬁbers. This is useful for short pulse delivery.

In the ﬁeld of semiconductor lasers, vertical-cavity

surface-emitting lasers (VCSELs) have developed as

a competitive alternative to the conventional edge-

emitting semiconductor lasers. The most recent advance

involves designs that have a horizontal laser cavity but

emits from the surface [71.53]. This combines the ease

of packaging of VCSELs with the high power and good

stability properties of edge-emitting lasers.

Several new designs of high power semiconductor

lasers that can be frequency shifted to the blue and green

spectral regions have been developed as rugged, efﬁcient

sources [71.54, 55]. One of these is a GaAs-based ver-

tical external cavity surface-emitting laser (VECSEL)

optically pumped with an 808 nm semiconductor laser.

This emits at 976 nm with a cavity that includes a wave-

length selector and a doubling crystal. The second

conﬁguration uses a semiconductor material as a gain

medium in an external cavity with a wavelength selec-

tor. In this case the frequency doubling crystal is outside

the cavity. These conﬁgurations have reached 20 mW cw

operation with high beam quality. One enabling tech-

nology is the use of microelectromechanical systems

(MEMS) for mirrors to tune VCELs [71.56].

New material conﬁgurations offer some advantages.

Photonic crystals can be used to produce nanocavity

lasers [71.57] while band-structure engineering can be

used to design quantum-cascade lasers [71.58]. The

latter are multiple-quantum-well heterostructures based

on intersub-band transitions. They operate in the infrared

to terahertz spectral region.

The broad spectral band available for laser gain is

the major distinguishing feature of liquid organic dye

lasers. This provides the ability to have tunable output

over a broad range of wavelength, the ability to gen-

erate ultrashort pulses, narrow linewidth cw operation,

and high average power operation. This variety of oper-

ating parameters keeps this class of lasers competitive

in a variety of applications. They have been especially

useful for laser spectroscopy in the visible region of the

spectrum and in the area of laser cooling. They have

achieved energies of up to 800 J per pulse and average

powers greater than 1 kW. The discovery of highly sta-

ble water-soluble dyes is an important advancement in

this ﬁeld [71.59].

Free-electron lasers (FELs) can produce coherent

emission at a wide range of wavelengths. Output for-

mats include ultrashort pulses and high powers [71.60].

Using a photocathode electron gun in a single pass, self-

ampliﬁed spontaneous emission mode, FELs can operate

at wavelengths where there are no mirrors with high re-

ﬂectivity. This type of laser emission has been extended

to the vacuum ultraviolet and the hard X-ray regions.

The technique of using a subharmonic seed laser has

been developed to improve the spectral purity of FEL

emission. The use of an energy-recovering accelerator

produces FELs with high average power (over 1 MW).

In the spectral region around 1 mm, FELs have produced

over 300 W of picosecond pulses.

Laser systems using various nonlinear optics tech-

niques continue to be developed. There has been

signiﬁcant research on high intensity, ultrashort pulse

lasers because of their special characteristics for atmo-

spheric propagation. Using femtosecond pulses above

a critical power level produces “light strings” that prop-

agate without dispersion for many kilometers due to

the balance between Kerr self-focusing and air ioniza-

tion [71.61]. Stimulated Brillouin scattering in water has

been shown to be effective in pulse compression and the

production of nondiffracting laser beams [71.62].

Part F 71.6

1048 Part F Quantum Optics

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Types of Lasers References 1049

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Part F 71

1051

Nonlinear Opt

72. Nonlinear Optics

Nonlinear optics is concerned with the propagation

of intense beams of light through a material

system. The optical properties of the medium can

be modiﬁed by the intense light beam, leading to

new processes that would not occur in a material

that responded linearly to an applied optical ﬁeld.

These processes can lead to the modiﬁcation of

the spectral, spatial, or polarization properties of

the light beam, or the creation of new frequency

components. More complete accounts of nonlinear

optics including the origin of optical nonlinearities

can be found in references [72.1–4].

Both the Gaussian and MKS system of units are

commonly used in nonlinear optics. Thus, we have

chosen to express the equations in this chapter

in both the Gaussian and MKS systems. Each

equation can be interpreted in the MKS system

as written or in the Gaussian system by omitting

the prefactors (e.g., 1/4πε

) that appear in square

brackets at the beginning of the expression on the

right-hand-side of the equation.

72.1 Nonlinear Susceptibility .......................1051

72.1.1 Tensor Properties ......................1052

72.1.2 Nonlinear Refractive Index ......... 1052

72.1.3 Quantum Mechanical Expression

for χ

(n)

....................................1052

72.1.4 The Hyperpolarizability ..............1053

72.2 Wave Equation in Nonlinear Optics........1054

72.2.1 Coupled-Amplitude Equations ....1054

72.2.2 Phase Matching ........................1054

72.2.3 Manley–Rowe Relations............. 1055

72.2.4 Pulse Propagation .....................1055

72.3 Second-Order Processes ....................... 1056

72.3.1 Sum Frequency Generation......... 1056

72.3.2 Second Harmonic Generation .....1056

72.3.3 Difference

Frequency Generation ............... 1056

72.3.4 Parametric Ampliﬁcation

and Oscillation.......................... 1056

72.3.5 Focused Beams .........................1056

72.4 Third-Order Processes .......................... 1057

72.4.1 Third-Harmonic Generation .......1057

72.4.2 Self-Phase and Cross-Phase

Modulation ..............................1057

72.4.3 Four-Wave Mixing ..................... 1058

72.4.4 Self-Focusing and Self-Trapping . 1058

72.4.5 Saturable Absorption .................1058

72.4.6 Two-Photon Absorption ............. 1058

72.4.7 Nonlinear Ellipse Rotation .......... 1059

72.5 Stimulated Light Scattering ..................1059

72.5.1 Stimulated Raman Scattering......1059

72.5.2 Stimulated Brillouin Scattering ...1060

72.6 Other Nonlinear Optical Processes ......... 1061

72.6.1 High-Order Harmonic Generation 1061

72.6.2 Electro-Optic Effect....................1061

72.6.3 Photorefractive Effect ................ 1061

72.6.4 Ultrafast and Intense-Field

Nonlinear Optics .......................1062

References .................................................. 1062

72.1 Nonlinear Susceptibility

In linear optics it is customary to describe the response

of a material in terms of a macroscopic polarization

(i. e., dipole moment per unit volume) which is linearly

related to the applied electric ﬁeld

E through the linear

susceptibility χ

(1)

. In order to extend the relationship

between

P and

E into the nonlinear regime, the polar-

ization is expanded in a power series of the electric ﬁeld

strength. We express this relationship mathematically by

ﬁrst decomposing the ﬁeld and the polarization into their

frequency components such that

E(r, t) =



E(r,ω

−iω

, (72.1)

P(r, t) =



P(r,ω

−iω

, (72.2)

where the summations are performed over both positive

and negative frequencies. The reality of

E and

P is then

Part F 72

1052 Part F Quantum Optics

assured by requiring that E(r,ω

) = E

∗

(r, −ω

) and

P(r,ω

) = P

∗

(r, −ω

). In this case the general expres-

sion for the Cartesian component i of the polarization at

frequency ω

is given by

(ω

) =[ε

]





(1)

(ω

)



(mn)

(2)

ijk

(ω

;ω

,ω

)

× E

(ω

)



jkl



(mno)

(3)

ijkl

(ω

;ω

,ω

(ω

)

× E

(ω

)

+···



(72.3)

where ijkl refer to ﬁeld components, and the notation

(mn), for example, indicates that the summation over

n and m should be performed such that ω

= ω

+ω

is held constant. Inspection of (72.3)showsthat

the χ

(n)

can be required to satisfy intrinsic permutation

symmetry, i. e., the Cartesian components and the cor-

responding frequency components [e.g., ( j,ω

) but not

(i,ω

)] associated with the applied ﬁelds may be per-

muted without changing the value of the susceptibility.

For example, for the second-order susceptibility,

(2)

ijk

(ω

;ω

,ω

) = χ

(2)

ikj

(ω

;ω

,ω

). (72.4)

If the medium is lossless at all the ﬁeld frequencies tak-

ing part in the nonlinear interaction, then the condition

of full permutation symmetry is necessarily valid. This

condition states that the pair of indices associated with

the Cartesian component and the frequency of the non-

linear polarization [i. e., (i,ω

)] may be permuted along

with the pairs associated with the applied ﬁeld compo-

nents. For example, for the second-order susceptibility,

this condition implies that

(2)

ijk

(ω

;ω

,ω

) = χ

(2)

kji

(−ω

;ω

, −ω

). (72.5)

If full permutation symmetry holds, and in addition all

the frequencies of interest are well below any of the

transition frequencies of the medium, the χ

(n)

are invari-

ant upon free permutation of all the Cartesian indices.

This condition is known as the Kleinman symmetry

condition.

72.1.1 Tensor Properties

The spatial symmetry properties of a material can be

used to predict the tensor nature of the nonlinear sus-

ceptibility. For example, for a material that possesses

inversion symmetry, all the elements of the even-

ordered susceptibilities must vanish (i. e., χ

(n)

= 0for

n even). The number of independent elements of the

nonlinear susceptibility for many materials can be sub-

stantially fewer than than the total number of elements.

For example, in general χ

(3)

consists of 81 elements,

but for the case of isotropic media such as gases,

liquids, and glasses, only 21 elements are nonvanish-

ing and only three of these are independent. The non-

vanishing elements consist of the following types: χ

(3)

iijj

(3)

ijij

,andχ

(3)

ijji

,wherei = j. In addition, it can be shown

that

(3)

iiii

= χ

(3)

iijj

+χ

(3)

ijij

+χ

(3)

ijji

. (72.6)

72.1.2 Nonlinear Refractive Index

For many materials, the refractive index n is intensity-

dependent such that

n = n

I , (72.7)

where n

is the linear refractive index, n

is the nonlinear

refractive index coefﬁcient , and I =[4πε

c|E|

/2π

is the intensity of the optical ﬁeld. For the case of

a single, linearly polarized light beam traveling in an

isotropic medium or along a crystal axis of a cubic

material, n

is related to χ

(3)



16π



12π

(3)

iiii

(ω;ω, ω, ω, −ω) .

(72.8)

For the common situation in which n

is measured in

units of cm

/Wandχ

(3)

is measured in Gaussian units,

the relation becomes





12π

×10

(3)

iiii

(ω;ω, ω, ω, −ω) .

(72.9)

There are various physical mechanisms that

can give rise to a nonlinear refractive index. For

the case of induced molecular orientation in CS

=3×10

−14

/W. If the contribution to the nonlin-

ear refractive index is electronic in nature (e.g., glass),

then n

≈ 2×10

−16

/W.

Part F 72.1

Nonlinear Optics 72.1 Nonlinear Susceptibility 1053

72.1.3 Quantum Mechanical Expression

for χ

(n)

The general quantum mechanical perturbation expres-

sion for the χ

(n)

in the nonresonant limit is (Under

conditions of resonant excitation, relaxation phenom-

ena must be included in the treatment, and the density

matrix formalism must be used [72.4]. The resulting

equation for the nonlinear susceptibility is then more

complicated)

(n)

···i

(ω

;ω

,... ,ω

)







···a

(g)

(ω

−ω

−···−ω

)

···µ

n- 1

n-1

(ω

−ω

−···−ω

) ···(ω

−ω

)

(72.10)

where ω

= ω

+···+ω

, N is the density of atoms

or molecules that compose the material, ρ

(g) is the

probability that the atomic or molecular population is

initially in the state g in thermal equilibrium, µ

the i

th Cartesian component of the (a

) dipole matrix

element, ω

is the transition frequency between the

states a

and g,andP

is the full permutation operator

which is deﬁned such that the expression that follows

it is to be summed over all permutations of the pairs

,ω

), (i

,ω

) ···(i

,ω

) and divided by the number

of permutations of the input frequencies. Thus the full

expression for χ

(2)

consists of six terms and that for χ

(3)

consists of 24 terms.

In the limit in which the frequencies of all the ﬁelds

are much smaller than any resonance frequency of the

medium, the value of χ

(n)

can be estimated to be

(n)





2µ



Nµ, (72.11)

where µ is a typical value for the dipole moment and

is a typical value of the transition frequency between

the ground state and the lowest-lying excited state. For

the case of χ

(3)

in Gaussian units, the predicted value

is χ

(3)

= 3×10

−14

, which is consistent with the meas-

ured values of many materials (e.g., glass) in which

the nonresonant electronic nonlinearity is the dominant

contribution.

72.1.4 The Hyperpolarizability

The nonlinear susceptibility relates the macroscopic po-

larization P to the electric ﬁeld strength E. A related

microscopic quantity is the hyperpolarizability, which

relates the dipole moment p induced in a given atom

or molecule to the electric ﬁeld E

loc

(the Lorentz local

ﬁeld) that acts on that atom or molecule. The relationship

between p and E

loc

(ω

)

=[ε

]





(ω

loc

(ω

)



(mn)

ijk

(ω

;ω

,ω

loc

(ω

loc

(ω

)



jkl



(mno)

ijkl

(ω

;ω

,ω

)

× E

loc

(ω

loc

(ω

loc

(ω

) +···



, (72.12)

where α

is the linear polarizability, β

ijk

is the

ﬁrst hyperpolarizability, and γ

ijkl

is the second hy-

perpolarizability. The nonlinear susceptibilities and

hyperpolarizabilities are related by the number density

of molecules N and by local-ﬁeld factors, which account

for the fact that the ﬁeld E

loc

that acts on a typical mol-

ecule is not in general equal to the macroscopic ﬁeld E.

Under many circumstances, it is adequate to relate E

loc

to E through use of the Lorentz approximation

loc

(ω) = E(ω) +



4πε



4π

P(ω) .

(72.13)

To a good approximation, one often needs to include

only the linear contribution to P(ω), and thus the local

electric ﬁeld becomes

loc

(ω) = L(ω)E(ω) , (72.14)

where L(ω) =



−1



ε(ω) +2



/3 is the local ﬁeld

correction factor and ε(ω) is the linear dielectric con-

stant. Since P(ω) = N p(ω),(72.3)and(72.12) through

(72.14) relate the χ

(n)

to the hyperpolarizabilities

through

(1)

(ω

) = L(ω

)Nα

(ω

), (72.15)

(2)

ijk

(ω

;ω

,ω

) = L(ω

)L(ω

)

× Nβ

ijk

(ω

;ω

,ω

(72.16)

(3)

ijk

(ω

;ω

,ω

) = L(ω

)L(ω

)

× Nγ

ijkl

(ω

;ω

,ω

(72.17)

For simplicity, the analysis above ignores the vector

character of the interacting ﬁelds in calculating L(ω).

A generalization that does include these effects is given

in [72.5].

Part F 72.1

1054 Part F Quantum Optics

72.2 Wave Equation in Nonlinear Optics

72.2.1 Coupled-Amplitude Equations

The propagation of light waves through a nonlinear

medium is described by the wave equation

∇

E −

∂

∂t

E =



4πε



4π

∂

∂t

P .

(72.18)

For the case in which

E and

P are given by (72.1),

the ﬁeld amplitudes associated with each frequency

component can be decomposed into their plane wave

components such that

E(r,ω

) =



(r,ω

·r

P(r,ω

) =



(r,ω

·r

, (72.19)

where k

= n(ω

)ω

/c is the magnitude of the wavevec-

tor k

. The amplitudes A

and P

are next decomposed

into vector components whose linear optical properties

are such that the polarization associated with them does

not change as the ﬁeld propagates through the material.

For example, for a uniaxial crystal these eigenpolariza-

tions could correspond to the ordinary and extraordinary

components. In order to describe the propagation and

the nonlinear coupling of these eigenpolarizations, the

vector ﬁeld amplitudes are expressed as

(r,ω

) =

(r,ω

) =

(r,ω

), (72.20)

where

is the unit vector associated with the

eigenpolarization of the spatial mode n at fre-

quency ω

. If the ﬁelds are assumed to travel along the

z-direction, and the slowly-varying amplitude approxi-

mation ∂

/∂z

 2k

∂A

/∂z is made, the change in

the amplitude of the ﬁeld as it propagates through the

nonlinear medium with no linear absorption is described

by the differential equation

(ω

)

=±



4πε



i2πω

n(ω

(ω

), (72.21)

where P

is the nonlinear contribution to the po-

larization amplitude P

, n(ω

) is the linear refractive

index at frequency ω

, and the plus (minus) sign indi-

cates propagation in the positive (negative) z-direction.

Sections 72.3 and 72.4 give expressions for the P

for various second- and third-order nonlinear optical

processes. Equation (72.21)) is used to determine the

set of coupled-amplitude equations describing a par-

ticular nonlinear process. For example, for the case of

sum-frequency generation , the two ﬁelds of frequency

and ω

are combined through second-order non-

linear interaction to create a third wave at frequency

=ω

+ω

. Assuming full permutation symmetry, the

amplitudes of the nonlinear polarization for each of the

waves are

(z,ω

) =[ε

]2χ

(2)

eff

A(z,ω

∗

(z,ω

−i∆kz

(72.22)

(z,ω

) =[ε

]2χ

(2)

eff

A(z,ω

∗

(z,ω

−i∆kz

(72.23)

(z,ω

) =[ε

]2χ

(2)

eff

A(z,ω

)A(z,ω

i∆kz

(72.24)

where ∆k = k

−k

is the wavevector mismatch

(see Sect. 72.2.2)andχ

(2)

eff

is given by

(2)

eff



ijk

(2)

ijk

(

∗

)

(

)

(

)

, (72.25)

where (

)

ı. For simplicity, the subscripts on each

of the ﬁeld amplitudes have been dropped, since only

one spatial mode at each frequency contributed. The

resulting coupled amplitude equations are

dA(ω

)



4π



i4πω

(2)

eff

n(ω

A(ω

∗

(ω

−i∆kz

(72.26)

dA(ω

)



4π



i4πω

(2)

eff

n(ω

A(ω

∗

(ω

−i∆kz

(72.27)

dA(ω

)



4π



i4πω

(2)

eff

n(ω

A(ω

)A(ω

i∆kz

(72.28)

72.2.2 Phase Matching

For many nonlinear optical processes (e.g., harmonic

generation) it is important to minimize the wave vec-

tor mismatch in order to maximize the efﬁciency. For

example, if the ﬁeld amplitudes A(ω

) and A(ω

) are

constant, the solution to (72.28) yields for the output

Part F 72.2

Nonlinear Optics 72.2 Wave Equation in Nonlinear Optics 1055

intensity

I(L,ω

) =



64π



32π



(2)

eff



I(ω

)I(ω

n(ω

)n(ω

×sinc

(∆kL/2), (72.29)

in terms of sinc x = (sin x )/x,whereI(L,ω

) =

(4πε

)n(ω

)c|A(L,ω

/2π,andI(ω

) and I(ω

) are

the corresponding input intensities. Clearly, the effect

of the wavevector mismatch is to reduce the efﬁciency

of the generation of the sum frequency wave. The

maximum propagation distance over which efﬁcient

nonlinear coupling can occur is given by the coherence

length

∆k

(72.30)

As a result of the dispersion in the linear refrac-

tive index that occurs in all materials, achieving phase

matching over typical interaction lengths (e.g., 5 mm) is

nontrivial. For the case in which the nonlinear mater-

ial is birefringent, it is sometimes possible to achieve

phase matching by insuring that the interacting waves

possess some suitable combination of ordinary and ex-

traordinary polarization. Other techniques for achieving

phase matching include quasiphase matching [72.5]and

the use of the mode dispersion in waveguides [72.6].

However, the phase matching condition is automat-

ically satisﬁed for certain nonlinear optical processes,

such as two-photon absorption (see Sect. 72.4.6)and

Stokes ampliﬁcation in stimulated Raman scattering

(see Sect. 72.5.1). One can tell when the phase match-

ing condition is automatically satisﬁed by examining the

frequencies that appear in the expression for the nonlin-

ear susceptibility. For a nonlinear susceptibility of the

sort χ

(3)

(ω

;ω

,ω

) the wave vector mismatch is

given in general by ∆k = k

−k

. Thus, for

the example of Stokes ampliﬁcation in stimulated Ra-

man scattering, the nonlinear susceptibility is given by

(3)

(ω

;ω

,ω

, −ω

) where ω

(ω

) is the frequency

of the pump (Stokes) wave, and consequently the wave

vector mismatch vanishes identically.

72.2.3 Manley–Rowe Relations

Under conditions of full permutation symmetry, there

is no ﬂow of power from the electromagnetic ﬁelds to

the medium, and thus the total power ﬂow of the ﬁelds

is conserved. The ﬂow of energy among the ﬁelds can

be described by the Manley–Rowe relations. For exam-

ple, for the case of sum-frequency generation, one can

deduce from (72.26, 27, 28)that



I(ω

)





I(ω

)



=−



I(ω

)



(72.31)

The expressions in square brackets are proportional to

the ﬂux of photons per unit area per unit time, and imply

that the creation of a photon at ω

must be accompanied

by the annihilation of photons at both ω

and ω

. Similar

relations can be formulated for other nonlinear optical

processes that are governed by a nonlinear susceptibil-

ity that satisﬁes full permutation symmetry. Since this

behavior occurs at the photon level, nonlinear optical

processes can lead to the generation of light ﬁelds that

have esoteric quantum statistical properties (Chapt. 78

and Chapt. 80).

A nonlinear optical process that satisﬁes the

Manley–Rowe relations is called a parametric process.

Conversely, a process for which ﬁeld energy is not

conserved, and thus Manley–Rowe relations cannot be

formulated, is said to be nonparametric. Thus, paramet-

ric processes are described by purely real χ

(n)

, whereas

nonparametric proceses are described by complex χ

(n)

72.2.4 Pulse Propagation

If the optical ﬁeld consists of ultrashort (<100 ps)

pulses, it is more convenient to work with the temporally

varying amplitude, rather than with the individual fre-

quency components. Thus, for a linearly polarized plane

wave pulse propagating along the z-axis, the ﬁeld is de-

composed into the product of a slowly varying amplitude

A(z, t) and a rapidly varying oscillatory term such that

E(r, t) = A(z, t) e

i(k

z−ω

+c.c., (72.32)

where k

/c. For a pulse propagating in a material

with an intensity-dependent refractive index, the prop-

agation can be described by the nonlinear Schrödinger

equation

∂A

∂z

iβ

∂

∂τ

= iγ |A|

A , (72.33)

where β



k/ dω



ω=ω

is the group velocity dis-

persion parameter, τ = t −z/v

is the local time for the

pulse, v

=[( dk/ dω)|

ω=ω

]

−1

is the group velocity, and

γ =[4πε

/2π is the nonlinear refractive index

parameter.

Part F 72.2

1056 Part F Quantum Optics

72.3 Second-Order Processes

Second-order nonlinear optical processes occur as

a consequence of the second term in expression

(72.3), i. e., processes whose strength is described by

(2)

(ω

;ω

,ω

). These processes entail the genera-

tion of a ﬁeld at frequency ω

= ω

+ω

in response

to applied ﬁelds at (positive and/or negative) frequen-

cies ω

and ω

. Several examples of such processes are

described in this Section.

72.3.1 Sum Frequency Generation

Sum frequency generation produces an output ﬁeld at

frequency ω

= ω

+ω

for ω

and ω

both positive. It

is useful, for example, for the generation of tunable radi-

ation in the uv if ω

and/or ω

are obtained from tunable

lasers in the visible range. Sum frequency generation is

described in detail in Sects. 72.2.1–72.2.3.

72.3.2 Second Harmonic Generation

Second harmonic generation is routinely used to convert

the output of a laser to a higher frequency. It is described

by χ

(2)

(2ω;ω, ω).Letη be the power conversion efﬁ-

ciency from frequency ω to 2ω. Assuming that phase

matching is perfect, and the pump wave at frequency ω

is undepleted by the interaction, a derivation analogous

to that for (72.29) yields

η = tanh



z/l



(72.34)

where the characteristic conversion length l is given by

l =



4π



√

n(ω)n(2ω)

4πωχ

(2)

(0)|

. (72.35)

Note that the conversion efﬁciency asymptotically ap-

proaches unity. In practice, conversion efﬁciencies

exceeding 80% can be achieved.

72.3.3 Difference Frequency Generation

Difference frequency generation can be used to create

light in the infrared and far infrared by generating the

difference frequency ω

= ω

−ω

(where ω

and ω

are positive and ω

>ω

) of two incident lasers. Con-

sider the case in which a strong (undepleted) pump

wave at frequency ω

and a weak (signal) wave

at ω

are incident on a nonlinear medium described

by χ

(2)

(ω

;ω

, −ω

) =χ

(2)

(ω

;ω

, −ω

). The ampli-

tude A(ω

) of the strong wave can be taken as a constant,

and thus the interaction can be described by ﬁnding si-

multaneous solutions to (72.26)and(72.27)forA(ω

)

and A(ω

). In the limit of perfect phase matching (i. e.,

∆k = 0), the solutions are

A(z,ω

) = A(0,ω

) cosh κz , (72.36)

A(z,ω

) = i



A(ω

)

|A(ω

∗

(0,ω

) sinh κz ,

(72.37)

where



16π



16π



(2)



|A(ω

. (72.38)

Equation (72.37) describes the spatial gro wth of the

difference frequency signal.

72.3.4 Parametric Ampliﬁcation

and Oscillation

For the foregoing case of a strong wave at frequency ω

and a weak wave with ω

<ω

incident on a second-

order nonlinear optical material, the lower frequency

input wave is ampliﬁed by the nonlinear interaction; this

process is known as parametric ampliﬁcation. Difference

frequency generation is a consequence of the Manley–

Rowe relations, as described above in Sect. 72.2.3.Since

= ω

+ω

, the annihilation of an ω

photon must be

accompanied by the simultaneous creation of photons

and ω

An optical parametric oscillator can be constructed

by placing the nonlinear optical material inside an op-

tical resonator that provides feedback at ω

and/or ω

When such a device is excited by a wave at ω

,it

can produce output frequencies ω

and ω

that sat-

isfy ω

+ω

= ω

. Optical parametric oscillators are

of considerable interest as sources of broadly tunable

radiation [72.7].

72.3.5 Focused Beams

For conceptual clarity, much of the discussion so far has

assumed that the interacting beams are plane waves.

In practice, the incident laser beams are often fo-

cused into the nonlinear material to increase the ﬁeld

strength within the interaction region and consequently

to increase the nonlinear response. However, it is un-

desirable to focus too tightly, because doing so leads

Part F 72.3